If your inner container is a solid box (i.e. rigid walls), then nothing will happen.
If the inner container is a balloon (interesting case), then it will shrink until the pressure of the balloon is equal to the pressure of the outer container plus an extra pressure necessary to stretch the balloon to the respective volume.
About the position of the inner container: it will fall as everything, except in the case that the buoyancy is stronger than the weight of the container:
in the case this condition is fulfilled, the inner container will go up.
In the case you have an equality between the two forces, then it will float.